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EUROPEAN ECONOMY

Economic and Financial Affairs

ISSN 2443-8022 (online)

EUROPEAN ECONOMY

Output Gap Estimation Using the European Union’s Commonly Agreed Methodology: Vade Mecum & Manual for the EUCAM SoftwareFrançois Blondeau, Christophe Planas and Alessandro Rossi

DISCUSSION PAPER 148 | OCTOBER 2021

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European Commission Directorate-General for Economic and Financial Affairs

Output Gap Estimation Using the European Union’s Commonly Agreed Methodology Vade Mecum & Manual for the EUCAM Software

François Blondeau, Christophe Planas and Alessandro Rossi

Abstract The EUCAM software is the officially validated tool for estimating potential growth and output gaps according to the European Union’s Commonly Agreed Methodology. This EUCAM methodology, which is comprehensively described in Havik et al. (2014), has been agreed between the EU’s Member States during discussions held at regular meetings of the Output Gap Working Group. The EUCAM software integrates all the operations that lead to the output gap and potential growth estimates, including data transformations, filtering, model estimation, and forecasts. Its user-friendly environment facilitates multi-country analyses and comparisons across different data vintages. It is open source and distributed together with the data files for EU countries at the publicly accessible repository circabc.europa.eu/ui/welcome (with users then prompted to follow the path “Browse categories - European Commission - Economic and Financial Affairs - Output Gaps - Library - PF method - EUCAM software”). The EUCAM software enables users to quickly replicate the output gap calculations made by the European Commission.

JEL Classification: C10, E60, C87. Keywords: Macroeconomic Policy, Econometric Methods, Computer Programs, Econometric Software. Acknowledgements: Werner Roeger, Valerie Vandermeulen, Rafal Raciborski and colleagues in Unit B3 of Directorate-General for Economic and Financial Affairs are gratefully acknowledged for their comments and suggestions. Closing date for this version was September 2021. Contact: François Blondeau, francois.blondeau@ec.europa.eu; Christophe Planas, christophe.planas @ec.europa.eu; Alessandro Rossi, alessandro.rossi1@ec.europa.eu.

EUROPEAN ECONOMY Discussion Paper 148

CONTENTS Executive summary ..................................................................................................................................... 2 1. Introduction ........................................................................................................................................ 4 2. Overview of the commonly agreed methodology ................................................................... 6

2.1. Potential output and output gap until T+2 ........................................................................................ 6 2.1.1. Labour ............................................................................................................................................................. 7 2.1.2. Capital .......................................................................................................................................................... 10 2.1.3. Total Factor Productivity ............................................................................................................................ 10

2.2. Medium-term forecasts: from T+3 to T+5 ......................................................................................... 12

2.2.1. Labour ........................................................................................................................................................... 12

2.3. Country specificities ............................................................................................................................ 13

2.3.1. Estonia ........................................................................................................................................................... 13 2.3.2. Ireland ........................................................................................................................................................... 13 2.3.3. Latvia ............................................................................................................................................................. 14 2.3.4. Luxembourg ................................................................................................................................................. 14

3. User manual ..................................................................................................................................... 16

3.1. Installation ............................................................................................................................................. 16 3.2. Getting started: the file EUCAMv10.xlsm ................................................................................................. 18

3.2.1. Worksheet Source ....................................................................................................................................... 18 3.2.2. Worksheet HP settings & series extensions ............................................................................................... 20 3.2.3. Worksheet TFP .............................................................................................................................................. 22 3.2.4. Worksheet NAWRU ...................................................................................................................................... 28 3.2.5. Worksheet Output Gap .............................................................................................................................. 33 3.2.6. Worksheet Manual ...................................................................................................................................... 37 3.2.7. Worksheet Upgrades .................................................................................................................................. 37

4. Data description ............................................................................................................................. 38

4.1. AMECO .................................................................................................................................................. 38 4.2. CUBS data ............................................................................................................................................. 41 4.3. Population ............................................................................................................................................. 41

5. Conclusion ...................................................................................................................................... 43 APPENDIX A AMECO data & acronyms ................................................................................................ 44 APPENDIX B Output files ............................................................................................................................ 49 APPENDIX C The Hodrick-Prescott filter ................................................................................................. 52 APPENDIX D The autoregressive model ................................................................................................. 52 REFERENCES ...................................................................................................................................... 53

Executive Summary

This discussion paper provides a detailed description of a new, user-friendly, soft-

ware tool aimed at simplifying the process of calculating potential output and output

gap estimates, using the EU′s commonly agreed, production function methodology

(henceforth referred to as the EUCAM i.e. the EU′s Commonly Agreed Methodology).

It also acts as a Vade Mecum for EUCAM since in addition to providing a user man-

ual for the software, it also includes a pedagogical summary of the key features of

the methodology, as well as a comprehensive description of all the data sources and

variables used in the calculations.

In terms of policy use, potential output and output gap estimates are essential

for assessing the productive (supply side) capacity and the cyclical (demand side)

positions of all of the EU′s economies. They are estimated by DG ECFIN, using the

transparent EUCAM rules as agreed by the Economic Policy Committee′s Output Gap

Working Group and the EPC and endorsed by the ECOFIN Council. The estimates

are applied in the framework of various economic and fiscal surveillance processes.

More specifically, the output gap indicator is used to cyclically adjust the budget

balances of the Member States (i.e. to produce their structural budget balances)

and the potential output indicator is used for assessing debt sustainability and for

comparing EU economies′ net government expenditures compared to their potential

growth rates (which is used as part of the expenditure benchmark criterion under the

preventive arm of the Stability and Growth Pact). In addition, potential output can

also be relevant for evaluating the sustainability and effectiveness of different structural

reform agendas, aimed at boosting the productive capacity of individual EU countries.

This paper has been written to complement the 2014 paper (Havik et al., 2014)

which currently provides the most comprehensive description of the EU′s official output

gap methodology. To put this new EUCAM software tool and the current paper in

their proper context, they are meant to be more practical and user-oriented, focusing

only on the essential information needed to compute the different indicators whilst

encouraging users to refer to the 2014 paper for a more exhaustive description of the

method and of the underlying economic theory.

The EUCAM software is an open source program which has been developed thanks

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to the collaboration of DG ECFIN and the Joint Research Centre. Its release is an

important step in enhancing the user-friendliness of the EU′s output gap estimation

methodology. The EUCAM software is used to calculate the official EC output gap and

potential output estimates since Spring 2021. It aims at streamlining the estimation

of the economic indicators used in the calculations and at making the process more

transparent. Regarding the first aim, it provides a common graphical user interface

which allows users to control each step of the calculation process. Regarding the

transparency goal, the EUCAM software relies entirely on open source technology

with publicly released code, allowing outside contributions to be considered in terms

of the future development of the software.

EUCAM is available on CIRCABC together with all of the data files (provided via

Excel files) and models needed to fully replicate the Commission′s latest output gap

calculations for each of the EU′s Member States, as well as for the euro area and EU

aggregates (using the estimates and forecasts for a wide range of economic variables

contained in either the Commission′s Spring or Autumn forecasting exercises). The

EUCAM software integrates the various programmes and data modules, which have

steadily grown in number over the years since the inception of the method in 2002,

with the result that all of the different components of the calculation process can now

be executed via one single graphical user interface. This interface offers a range of

user-friendly facilities including menu-driven input options; analytical tools for multi-

country analyses; as well as graphical tools for comparisons across countries or vis-a-vis

previous data vintages.

In terms of the structure of the paper, following a short introductory section, the

paper proceeds to discuss briefly the essential elements of EUCAM by (1) describing

the components of its production function approach (i.e. labour, capital and total

factor productivity); (2) the way these components are forecasted for the medium-

term projections (over a three to five year horizon); (3) and finally describes country

specificities that are economically justified exceptions to the methodology. The third

section discusses the EUCAM software, starting with the installation process, and

moving to operational instructions. This section describes the step-by-step functioning

of the software using both screen captures and text to guide the user through the

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process. The final section includes a comprehensive overview of the various data sources

used in the EUCAM calculations and of the specific data needed for the estimations.

This specific structure has been chosen since one of the main aims of this discussion

paper is to demonstrate that EUCAM is not as complex as many commentators have

suggested. Consequently, all the requisite elements needed to trace the final output

gap results back to the initial data inputs are made available in summary form. This is

achieved since, in addition to a description of the user interface and data inputs used for

the calculations, the paper also includes the key equations which ultimately determine

the output of the methodology and a description of the econometric methodology

implemented to estimate the pivotal NAWRU and potential TFP variables. In this

way, interested readers / users can discover that EUCAM is not a complex “black-box”

but in fact is a relatively simple, rules-based, and 100% transparent methodology where

the important data inputs, allied with the key mathematical equations underpinning

the methodology, are combined to produce the final potential output and output gap

estimates.

Finally, EUCAM should not be seen in static terms since there is a strong likelihood

that specific details of the approach will be amended by the EU′s Member States in

the years to come on the basis of the practical experience garnered from using the

methodology in the Commission′s various surveillance exercises. As the methodology

changes, updated versions of the software and of the current paper will be placed on

CIRCABC. The broad structure of the current paper, however, will be retained since

it tries to combine the benefits of having a Vade Mecum of the methodology itself,

allied to the practicality of a “hands-on” user manual for the software which is used

to replicate, in a logical step-by-step process, the potential output and output gap

calculations used for EU policy surveillance purposes.

1 Introduction

The EUCAM software enables the European Union′s Commonly Agreed Methodology to

estimate potential growth and output gaps. This methodology, which is described in Havik

et al. (2014), has been agreed between EU Member States during discussions held at regular

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meetings of the Output Gap Working Group (OGWG)1. It relies on the Cobb-Douglas

production function; a brief overview is given in Section 2. A user manual to the EUCAM

software is provided in Section 3. The software performs all the operations that lead to the

output gap and potential growth estimates:

• it prepares all the preliminary data transformations and forecast extensions;

• it applies the Hodrick-Prescott (HP) filter to detrend the participation rate and hours

worked;

• it runs Program GAP (Planas and Rossi, 2020) to get potential Total Factor Productiv-

ity (TFP) and the Non-Accelerating Wage inflation Rate of Unemployment (NAWRU);

• it puts all the intermediate results together to build the potential growth and output

gap estimates;

• and, given medium-term extensions of the working age population and the investment

to capital ratio, it elaborates five-year-ahead forecasts of potential growth.

Several facilities like menu-driven inputs, multi-country analysis, and graphical comparisons

using another data vintage are offered. The program is open-source and can be downloaded

in CIRCABC at

circabc.europa.eu/ui/welcome

following the path Browse categories - European Commission - Economic and Financial

Affairs - Output Gaps - Library - PF method - EUCAM software. Users are kindly invited

to read the EU Licence Agreement. The program runs on 64-bit computers equipped with

MS Windows 10. Installation is described in Section 3. For application to EU Member

States, the official data set currently used by the European Commission is also provided via

Excel files. All details about this data set are given in Section 4. Section 5 concludes.

1The OGWG members are gratefully acknowledged for their suggestions and software testing.

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2 Overview of the commonly agreed methodology

For the sake of completeness we give a brief overview of the EUCAM; interested readers

can find a more exhaustive description in Havik et al. (2014). Given yearly observations

until time T plus additional two-year-ahead forecasts for all input variables that come from

the European Commission′s Spring and Autumn Economic Forecasts, potential growth and

output gap are first estimated until time T+2 as explained in Section 2.1. These results are

then used to build medium-term forecasts from T+3 to T+5 using a simultaneous system

of equations which is discussed in Section 2.2.

2.1 Potential output and output gap until T+2

The EUCAM assumes that output Y is produced by labour L, capital K, and total factor

productivity TFP in a Cobb-Douglas technology as in:

Y = TFP Lα K1−α with α = 0.65 (2.1)

Potential output Y POT is determined by potential total factor productivity POTTFP ,

potential labour LP , and capital according to:

Y POT = POTTFP LPα K1−α (2.2)

The output gap Y GAP relates potential to GDP as in:

Y = Y POT (1 + Y GAP/100) (2.3)

where Y GAP is given in percentage points. Figure 1 gives an overview of the T+2 method-

ology, which is reproduced from Havik et al. (2014).

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Figure 1 Overview of the production function approach

Labour, capital, and TFP are described next.

2.1.1 Labour

Labour is measured in terms of hours as the product of hours worked per employee hperet

times the number of employees emplt:

Lt = hperet × emplt (2.4)

Potential labour LPt equals the product of the hours worked trend hperehpt times the

employment trend emplpt:

LPt = hperehpt × emplpt (2.5)

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The trend in hours worked is calculated using the HP filter with an inverse signal to noise

ratio λ set equal to 10, after extending the series from T+3 to T+8 using an autoregressive

model (see Appendix C & D):

heperehpt = HP (hperet, t = 1, · · · , T + 8, λ = 10) (2.6)

To obtain the employment trend, the total number of employees is first decomposed into the

working age population popwt, the participation rate partt, and the rate of employment, or

equivalently one minus the harmonized rate of unemployment 1− lurharmt/100:

emplt = popwt × (partt/100)× (1− lurharmt/100) (2.7)

where both the participation rate and the labour unemployment rate are in %. The potential

number of employees verifies:

emplpt = popwt × (partst/100)× (1− nawrut/100) (2.8)

where partst denotes the trend in participation rate and nawrut the non-accelerating wage

inflation rate of unemployment. The trend in participation rate is calculated using the HP

filter with an inverse signal to noise ratio λ set equal to 10, after extending the series from

T+3 to T+8 using an autoregressive model:

partst = HP (partt, t = 1, · · · , T + 8, λ = 10) (2.9)

where the participation rate is obtained by inverting (2.7).

The trend unemployment rate is calculated as an anchored NAWRU (Hristov et al., 2017).

In a first step the unemployment rate is decomposed in a trend (i.e. NAWRU) and a cycle

which are assumed to evolve stochastically as in:

lurharmt = nawrut + ct

∆nawrut = ηt−1 + ant

ηt = ηt−1 + aηt

ct = φc1ct−1 + φc2ct−2 + act (2.10)

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where ∆ denotes first-difference. The trend component (nawrut) follows a second-order

integrated process without level shocks in all EU Member States, i.e. V (ant) = 0, except for

Slovenia for which a random walk without drift is assumed, i.e. V (aηt) = 0. The cyclical

component of unemployment is related to a labour cost indicator wt through the Phillips

curve equation:

wt = µw + φw1wt−1 + β0ct + β1ct−1 + γ′w∆2zt + awt (2.11)

All shocks ant, aηt, act, and awt are independent and normally distributed white noises. The

vector ∆2zt contain the second difference of exogenous variables such as labour productivity,

terms-of-trade, and the wage share, whose measurement is explained in Section 4.

The Phillips curve (2.11) is compatible with both backward- and forward-looking expec-

tations: with backward-looking expectations the labour cost indicator corresponds to the

change in wage inflation whereas with forward-looking expectations the relevant indicator

is the growth of real unit labour cost - see Section 4. No exogenous information is used

in the forward-looking Phillips curve. Instead, the forward-looking assumption leads to the

restriction β1 = β0(φw1 − 0.99)/(0.99φw1 − 1)φc2, where 0.99 is the assumed discount factor.

Model (2.10)-(2.11) is estimated by maximum likelihood after casting the model equations

in a state space format and launching the Kalman recursions from a diffuse initialisation;

details can be found in the GAP Manual.

In a second step the NAWRU is corrected to incorporate an anchor St to which it is

assumed to converge at some horizon T + h under the hypothesis of no-policy change; in

short it is assumed that nawruT+h = ST+h, where h is typically equal to ten years. The

anchor is built using a panel regression on structural indicators of the labour market (Hristov

and Roeger, 2020).

Plugging the NAWRU estimates together with the working age population and the smoothed

participation rate into (2.5) yields potential labour up to T+2.

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2.1.2 Capital

The net capital stock is calculated using the Perpetual Inventory Method. Capital stock

grows through real gross fixed capital formation IQ and it decreases with the consumption

of capital CFC. Given the investment deflator defl and a starting point in 1960, real capital

stock K is measured as:

Kt = Kt−1 + IQt −CFCtdeflt

(2.12)

Two-year-ahead forecasts are provided by the country-desks in DG ECFIN. Capital con-

tributes entirely to potential output and it is not subject to any transformation in the

EUCAM.

2.1.3 Total Factor Productivity

TFP is calculated by exponentiating the Solow residual srt defined by:

srt = log Yt − α logLt − (1− α) logKt (2.13)

Two-year-ahead forecasts of the Solow residual are provided by the country-desks in DG

ECFIN as part of the European Economic Forecasts. The Solow residual including the

two-year-ahead extension is then decomposed into a trend srkt plus a cycle ct using the

model:

srt = srkt + ct

∆srkt = µp + ηt−1 + apt

ηt = ρηt−1 + aηt

ct = 2 Acos(2π/τ)ct−1 − A2ct−2 + act (2.14)

The trend component srkt follows a damped trend model with average growth µp. The

cyclical component ct follows instead a second-order autoregressive process which is parame-

terized in terms of amplitude A and periodicity τ . The productivity cycle also appears in the

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fluctuations of capacity utilization, cubst, so the model is completed with the measurement

equation:

cubst = µcu + βcuct + et

et = φcuet−1 + acut (2.15)

where et is an unobserved idiosyncratic component which can be autocorrelated. All shocks,

namely apt, aηt, act, and aet are independent and normally distributed white noises. The cubs

indicator is built from capacity utilization surveys in industry and two economic sentiment

indicators for construction and services - see Section 4. Contrary to the Solow residual, no

T+2 forecasts are available for cubs which ends in time T or sometimes earlier. To cope

with this the estimation procedure allows for missing observations.

Model (2.14)-(2.15) is estimated in the Bayesian framework given prior distributions for

each EU Member State. Given the prior distributions, the Solow residual, and the capacity

utilization series, the trend component srkt is estimated by its posterior mean in a Bayesian

inference as explained in the GAP Manual. The estimate of potential TFP is then obtained

as:

POTTFPt = exp(srkt) (2.16)

Plugging potential TFP (2.16), potential labour (2.5), and capital into (2.2) yields potential

GDP up to T+2 together with the output gap.

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2.2 Medium-term forecasts: from T+3 to T+5

Potential output and capital are jointly forecast from T+3 to T+5 using the following system

of non-linear equations:

Y POTt = exp(srkt) LPαt K1−α

t

Kt = IQt + (1− δt) Kt−1

IQt = (IY POTt/100) Y POTt

Yt = Y POTt (1 + Y GAPt/100) (2.17)

Given inputs from srkt, LPt, δt, IY POTt, Y GAPt, the model above yields medium-term

projections for Y POTt, IQt, Kt, Yt, and srt, using the recursive method discussed in Juillard

(1996). The output gap which is available until T+2 is extended to T+5 by assuming that

it closes in three years as in:

Y GAPT+3 =2

3Y GAPT+2

Y GAPT+4 =1

3Y GAPT+2

Y GAPT+5 = 0 (2.18)

Investment to potential ratio IY POTt is extended to T+5 using an autoregressive model.

The other inputs are described below.

2.2.1 Labour

Potential labour in (2.5)-(2.8) depends on the working age population, on the trend par-

ticipation rate, and on the NAWRU. The working age population is available until T+2.

Predictions until T+5 are built using the long-range projections of the 15-74 population by

Eurostat (POPAF) as in:

popwt = popwt−1 ×POPAFt−1 + POPAFtPOPAFt + POPAFt+1

(2.19)

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The NAWRU is extended from T+3 to T+5 according to:

nawruT+3 = nawruT+2 +1

2(nawruT+2 − nawruT+1)

nawruT+4 = nawruT+3

nawruT+5 = nawruT+4 (2.20)

Both the trend of hours worked per employee in (2.6) and the trend participation rate in

(2.9) are available until T+5. Plugging them together with the population forecasts (2.19)

and the nawru extension (2.20) into (2.5)-(2.8) yields potential labour until T+5.

2.3 Country specificities

On 4 September 2017, the Economic and Financial Committee approved principles and gov-

ernance rules for making economically justified country-specific exceptions to the EUCAM.

The amendments adopted to date are described below.

2.3.1 Estonia

Although for Estonia the CUBS series is available since 1995, it has been decided to take

2002 as starting year like for the other Baltic countries. The change was endorsed at the

December 2020 meeting of the OGWG and is in force since the Spring 2021 forecast.

2.3.2 Ireland

The capital stock series for Ireland is adjusted in order to correct an anomaly in the national

accounts in this country. The anomaly is caused by a relatively small number of firms that

relocated intellectual property assets to this country: it happens that the relocations increase

capital stock but leave gross fixed capital formation unaffected, thus breaking the Perpetual

Inventory formula (2.12). To remedy the issue, the EPC agreed on a first set of adjustments

on 20 October 2016. The OGWG suggested further refinements which were adopted by the

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EPC on 15 October 2018. Capital stock for Ireland KIEt is now calculated as follows:

until 2014: KIEt = Kt

2015: KIEt = 1.53×Kt

2016: KIEt = KIE

t−1 + IQt −CFCtdeflt

2017: KIEt = KIE

t−1(1− δ) (2.21)

where Kt is the capital stock obtained from the Perpetual Inventory Method as described in

Section 2.1.2, and δ is the implied capital depreciation rate between 2015 and 2016.

2.3.3 Latvia

A measurement issue with the consumption of fixed capital distorts the calculation of the

depreciation rate for Latvia. The OGWG proposed to adjust the depreciation rate for Latvia

to more closely resemble that of Estonia and Lithuania. The Economic Policy Committee

(EPC) endorsed this exception on 23 January 2018. The depreciation rate for Latvia is

calculated as the arithmetic mean of depreciation rate of Estonia and Lithuania.

2.3.4 Luxembourg

In Luxembourg, cross-border employment amounts to 40% of total employment, pulling the

participation rate (i.e. the ratio of employment to working age population) above 100%. To

address this issue, the OGWG agreed to model domestic and cross-border workers separately

in the calculation of the participation rate for Luxembourg. The EPC endorsed this exception

by January 2018. The measurement of labour in Luxembourg is amended as follows:

Lt = emplnt × hperent + emplcbt × hperecbt

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where the superscripts ’n’ and ’cb’ refer to national and cross-border respectively. Expanding

employment in the two groups yields:

Lt = popwnt × (partnt /100)× (1− lurharmnt /100)× hperent +

+ popwcbt × (partcbt /100)× (1− lurharmcbt /100)× hperecbt

Contrary to the national group, for the cross-border workers no information is available on

the participation rate, the unemployment rate, and on the pool of working age population.

Assuming a participation rate equal to 100% and no unemployment, the cross-border working

age population amounts to cross-border employment. Assuming further that the cross-

borders work on average the same number of hours as the nationals, total hours worked

verifies:

Lt = popwnt × (partnt /100)× (1− lurharmnt /100)× hperet + emplcbt × hperet

(2.22)

Equation (2.22) measures labour for Luxembourg. Potential labour is derived similarly,

except that the trend of cross-border employment is needed. Rather than calculating trend

cross-border employment directly, the trend of the ratio of cross-border employment to total

employment is calculated via HP-filtering with inverse signal to noise ratio set equal to 10,

after extending the series with forecasts from an ARI(1,1) model. Eventually, potential

labour in Luxembourg is obtained by solving:

LPt = popwnt × (partsnt /100)× (1− nawrunt /100)× hperehpt

+ (emplcbtemplt

)hp × LPt

where the number of cross-border workers emplcbt is measured as the difference between

domestic and national employment:

emplcbt = emplt − emplnt= NETD −NETN using AMECO labels. (2.23)

The correction implicitly yields a participation rate equal to 60% in 2017 instead of more

than 100% when no distinction is made between national and cross-borders workers.

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3 User manual

3.1 Installation

Installation is made by running the self-extracting executable file EUCAMv10setup.exe. By

default the program is installed in the folder C:\EUCAMv10 but the users can change the

location. Unless a recent version of Matlab such as 2018b is already available on the disk,

users are proposed to install Matlab Runtime routines. These routines serve to visualise and

to tune the prior distributions for the parameters of the TFP-CU model (2.14)-(2.15). This

step can be skipped by simply answering ’No’ if the user is not interested in these aspects.

Once completed, the installation yields the files and sub-folders shown in Figure 2:

Figure 2 EUCAM Main Folder

The folder structure in Figure 2 is explained below:

– Year Season F: contains the data files, the models, and the results for final version

F of data vintage Year in period Season. Season is typically Spring or Autumn, and

sometimes Winter. For instance 2020 Spring F refers to the final version of data vintage

Spring 2020.

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– bin: contains the Python executable EUCAMv10.exe, the files related to the graphical

interface, and the dynamic link library version of Program GAP, gap50dll.dll.

– log: contains log files written during program execution.

– src: contains the Python source code.

– upgrades: EUCAM can be easily updated by importing the latest program files *.bas,

*.dll, and *.exe that are stored in this folder - see subsection 3.2.7 ’Upgrades’.

The main folder contains the following files:

– The excel interface EUCAMv10.xlsm.

– The licence agreements EUCAMLicence.txt and GAPLicence.txt that the users are

kindly invited to read.

– The EUCAM user-guide and the GAP manual where details about the econometric

methodology implemented to estimate the NAWRU and potential TFP can be found.

The software requires that the decimal character is set to a dot as in 1.234 and not as a

comma - please check your regional settings following the path Region - Additional Settings

in the Windows Control Panel.

The Excel interface reads and writes namelist files (*.nml) which are given as input to GAP.

Each namelist file contains model equations and all necessary statistical information in a

human-readable format. To ease reading, it is recommended to associate the *.nml filetype

to a preferred text editor. In Windows 10, this can be managed in the Settings area, following

the path Default Apps - Choose default apps by file type.

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3.2 Getting started: the file EUCAMv10.xlsm

To start the user needs to open the Excel interface EUCAMv10.xlsm. Seven worksheets are

available. They are described below.

3.2.1 Worksheet Source

Figure 3 EUCAM Worksheet Source

In the worksheet Source, the user can set the :

– Name of the current vintage: the format is Year Season F, referring to the Final

storage of data in Season typically Spring, Autumn, or Winter of vintage Year. All

inputs and outputs related to this vintage are located in the directory named according

to this name.

– Last year of data including the short-term forecasts: for instance, 2022 for

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vintage Autumn 2020 since it is extended with two years of forecasts.

– Name of the previous vintage: the previous vintage to be used for comparisons,

written in the same format as for current vintage. All input and output files related

to this vintage are located in the directory named accordingly.

– Last year of data including the short-term forecasts in the previous vintage:

for instance, 2021 for vintage Spring 2020 which is extended with two years of forecasts

starting from 2019.

– Ameco data file The series in the Ameco data file are described in Section 4. A link

to the equations in Section 2 is provided.

– CUBS data The capacity utilization series in the CUBS data file are described in

Section 4.

– Population data The series in the Population data file are described in Section 4.

The data files in the last three items must be located in the folder corresponding to the

current vintage.

19

3.2.2 Worksheet HP settings & series extensions

The HP-filter is used to estimate the trend of the participation rate and of hours worked

until T+5. The investment to potential output ratio is instead extended to T+5 using

autoregressive forecasts. The worksheet is illustrated in Figure 4.

Figure 4 EUCAM Worksheet HP settings & series extensions

The worksheet is organized as follows:

– Participation rate: the user can set the autoregressive order AR Order, include

a Constant or a linear time trend in the model used for forecast three additional

years, and select the AR Starting year of the sample used to estimate the parameters.

The inverse signal to noise ratio Lambda for the HP filter can be inserted in Column

F, and select the HP Starting year for computing the filtered estimates.

– Hours worked: the user can set the autoregressive order AR Order, include or not a

20

Constant, select the AR starting year of the sample used to estimate the param-

eters, insert the inverse to noise ratio Lambda, and select the HP Starting year for

computing the filtered estimates.

– Investment to potential output ratio: the user can set the autoregressive order

AR order, include a Constant, and select the AR starting year for model estimation.

– TFP: the user can enter in columns S-X the settings for HP filtering as an alternative

to the EUCAM for decomposing TFP.

– NAWRU: the user can enter in columns Z-AD the settings for HP filtering as an alternative

to the EUCAM for estimating the NAWRU. This is currently used in the case of IE.

The constant used for uncentering the NAWRU as discussed in Section 2.4.2 of Havik

et al. (2014) appears in column AE entitled ’Adjustment factor’.

21

3.2.3 Worksheet TFP

The worksheet TFP sets up the estimation of potential TFP and TFP gap up to T+2.

Figure 5 EUCAM Worksheet TFP

– Selection: the user can include or exclude a country from the TFP decomposition

process by setting Selection = Y or N. If N is selected so that no TFP decomposi-

tion is performed, the TFP trend must be provided in an Excel file with the vintage

information in its name, for instance tfp 2019 Autumn F.xls in the format shown in

Figure 6, otherwise no output gap calculation can be made for this country.

22

Figure 6 The data file tfp 2019 Autumn F.xls

– Method: either CAM or HP.

– Press View to visualise the model file which contains all information about the model

specification. For instance AT TFP 2020 Autumn F m0.nml contains a model specifica-

tion for Austria. The namelist syntax is described in the GAP manual.

– Press Choose to pick a model file located in the vintage folder: The selected file appears

in column D.

– Press Priors2 to revise the prior distributions. The current model is left unchanged.

The user is first asked to choose the name of the file which will contain the new

2This facility is available if the user allowed the Matlab Runtime routines to be installed running the

program setup.

23

priors. Once entered, an interface where the prior distributions can be tuned through

the menu-entry Priors appears as shown in Figure 7. Please notice that the interface

can take several minutes to load the first time it is launched. Further details on this

interface can be found in the GAP Manual.

Figure 7 EUCAM Worksheet TFP - Interface for updating priors

– Press Model to revise the model specification. This action opens the Specifications

worksheet. The user can choose a model name, change the model specifications, update

the bounds, and save the new model file by pressing Update. The interface for setting

priors displayed in Figure 7 opens automatically.

24

Figure 8 EUCAM Worskheet TFP - Interface for updating the model

– Press RUN to estimate potential TFP for each country for which estimation is set to

’Y’.

– Press Charts to see graphics. The Chart button opens the file Country TFP Vintage

Modelname.pdf located in the vintage folder. Eight panels are displayed:

– the TFP series in the current and previous vintage plus the respective trends;

– the growth rate of TFP and trend TFP in the current and previous vintage;

– as above with the addition of the HP trend;

– the CUBS series together with its cyclical part which is commonly shared with

TFP;

– the revisions to the TFP cycle compared to the previous vintage;

25

– the revisions to TFP, TFP trend, and CUBS compared to the previous vintage.

The charts focus on the last fourty years. An example is given in Figure 9.

26

Figure 9 EUCAM Worskheet TFP - Charts

1983 1989 1995 2001 2007 2013 2019 2025-10.28

-6.31

-2.33

1.64

5.62CUBS

1983 1989 1995 2001 2007 2013 2019 2025-9.55

-5.76

-1.98

1.80

5.58CUBS (- -) and TFP gap

1983 1989 1995 2001 2007 2013 2019 2025-7.27

-7.16

-7.05

-6.94

-6.83TFP (- -) and trend TFP

1983 1989 1995 2001 2007 2013 2019 2025-0.04

-0.02

-0.00

0.01

0.03TFP growth (- -) and TFP trend growth rate

1983 1989 1995 2001 2007 2013 2019 2025-7.26

-7.15

-7.05

-6.94

-6.83TFP (- -), trend TFP, and HP trend TFP (.)

1983 1989 1995 2001 2007 2013 2019 2025-0.03

-0.02

-0.00

0.02

0.03TFP growth (- -), trend TFP growth and HP trend TFP growth (.)

1983 1989 1995 2001 2007 2013 2019 2025-0.07

0.17

0.41

0.65

0.89TFP gap: 2020_AUTUMN_F minus 2020_SPRING_F

1983 1989 1995 2001 2007 2013 2019 2025-0.01

-0.00

0.00

0.01

0.02TFP, trend TFP and CUBS: 2020_AUTUMN_F minus 2020_SPRING_F

TFP Trend TFP CUBS

TFP report for AT, 2020_AUTUMN_F 2020_SPRING_F vs

Statistics for 2020_AUTUMN_F :. Beta coef. = 1.71. Cor. TFP gap/CU : 0.84Statistics for 2020_SPRING_F :. Beta coef. = 1.6. Cor. TFP gap/CU : 0.8

EUCAM v1 (2021-04-06 10:39:59)

– Press Posterior to see the posterior distribution of the estimated parameters, together

with the relative prior distribution.

27

3.2.4 Worksheet NAWRU

The worksheet NAWRU controls the NAWRU estimation up to T+2.

Figure 10 EUCAM Worksheet NAWRU

– Selection the user can include or exclude a country from the NAWRU estimation

process by setting Selection=Y or N. If N is selected so that no estimation is performed,

the NAWRU must be provided in an Excel file with the vintage information in its name,

for instance NAWRU 2020 Autumn F.xls in the format shown in Figure 11, otherwise no

output gap calculation can be made for this country.

28

Figure 11 The data file NAWRU 2020 Autumn F.xls

– Method either CAM or HP.

– Press View to visualise the model file which contains all information about the model

specification.

– Press Choose to pick a model file located in the vintage folder: for instance AT NAWRU

2020 Autumn F mf.nml contains a model specification for Austria. The selected file

appears in column D as shown in Figure 10.

– Press Bounds to revise the bounds of the variance parameters - see model (2.10)-(2.11).

Use can be made of the information displayed in columns M-N, P-Q, and S-T, where

the current variance bounds are reported. The user is first asked to choose the name

of the model file which will contain the new bounds.

29

– Press Model to revise the model specification. This action opens the “Specifications”

worksheet as shown in Figure 12. The user can choose a model name, change the

model specifications, and save the new model file by pressing Update.

Figure 12 EUCAM Worskheet NAWRU - Interface for updating the model

– Press RUN to estimate the NAWRU for each country for which estimation is set to ’Y’.

– Press Charts in column View to see graphics as illustrated in Figure 13. Six panels

are displayed:

– The top-left panel displays the unemployment rate and the NAWRU in the current

and previous vintage;

– The top-right panel shows the unemployment gap in the last two vintages.

30

– The mid-left panel shows the labour cost indicator in the Phillips curve (2.11),

change in wage inflation in the case of AT, together with its ’fitted’ component

which refers to its explained component, i.e. the component of the labour cost

indicator which is common to unemployment. The fitted values obtained by

applying the previously used model to the new vintage is also displayed in dots -

label ’wpm’.

– The mid-right panel shows the revisions to the unemployment gap compared to

the previous vintage in the last years.

– The bottom-left panel displays the revisions to the unemployment rate, the NAWRU,

and the wage indicator compared to the previous vintage in the last years.

– The bottom-right panel shows the unemployment gap times its coefficient (β < 0)

in (2.11) together with the labour cost indicator.

31

Figure 13 EUCAM Worskheet NAWRU - Charts

2001 2004 2007 2010 2013 2016 2019 2022-2.41

-1.36

-0.31

0.74

1.78Wage indicator: actual (- -), fitted, fitted wpm (.)

2001 2004 2007 2010 2013 2016 2019 20223.38

4.06

4.75

5.44

6.12Unemployment rate (- -) and NAWRU

2001 2004 2007 2010 2013 2016 2019 2022-1.22

-0.76

-0.30

0.17

0.63UR, NAWRU, and WI: 2020_AUTUMN_F minus 2020_SPRING_F

UR NAWRU Wage indicator

2001 2004 2007 2010 2013 2016 2019 2022-0.63

-0.18

0.26

0.71

1.15Unemployment gap

2001 2004 2007 2010 2013 2016 2019 2022-0.32

-0.18

-0.04

0.10

0.24Unemployment gap: 2020_AUTUMN_F minus 2020_SPRING_F

2001 2004 2007 2010 2013 2016 2019 2022-4.18

-1.89

0.40

2.69

4.98-Unemp. gap, wage inf. (- -), and real unit labor cost (.)

NAWRU report for AT, 2020_AUTUMN_F 2020_SPRING_F vs

2020_AUTUMN_F : Beta t-stat = -1.47. LB UBSlope var. 0.01 0.1Trend var. 1e-08 0.05Cycle var. 0.07 0.152020_SPRING_F : Beta t-stat = -1.64. LB UBSlope var. 0.01 0.1Trend var. 1e-08 0.05Cycle var. 0.07 0.15

EUCAM v1 (2021-04-29 09:53:15)

– Press Results to see the SOL.txt file that contains the maximum likelihood parameter

estimates. The estimated value of a selection of parameters is reported from columns

L onward.

32

3.2.5 Worksheet Output Gap

The worksheet Output Gap displays the Potential Growth and Output Gap estimates.

Figure 14 EUCAM Worskheet Output Gap

33

– Press RUN to build the new Output Gap estimates. Depending on which box is ticked,

OG only or OG, TFP, NAWRU, also the potential TFP and the NAWRU are recalcu-

lated, otherwise used is made of the last estimates. Columns E-H displays potential

growth from T-1 to T+2 and columns I-L the respective output gap estimates. Columns

N to U reports the values obtained with the previous vintage.

– Press Charts to see graphics as illustrated in Figure 15. Twelve panels are displayed:

– The first-left panel shows the output gap in the current and previous vintages.

– The first-right panel shows the revisions in output gap between the current and

the previous vintage.

– The second-left panel shows GDP and potential GDP in the current against the

previous vintage.

– The second-right panel shows investment to potential for the current and previous

vintage.

– The third-left panel displays the current and previous vintages of TFP and trend

TFP.

– The third-right panel shows Unemployment and NAWRU in the current against

the previous vintage.

– The fourth-left panel shows the participation rate and its trend in the current and

previous vintages.

– The fourth-right panel shows the revisions in the current vintage of participation

rate and trend participation rate.

– The fifth-left panel shows average hours worked and its trend in the current and

previous vintages.

– The fifth-right panel shows the revisions in the current vintage of hours worked.

– The sixth-left panel shows population of working age in the current and previous

vintages.

34

– The sixth-right panel shows the revisions in the current current vintage of popu-

lation of working age.

– Press Load configuration to load the settings saved in a previous work session,

for instance with a different data vintage. The configuration is written in the file

Configurationfile.nml which contains all the settings introduced in the Excel inter-

face. The program overwrites the file Configurationfile.nml at each execution.

35

Figure 15 EUCAM Worskheet Output Gap - Charts

2017 2019 2021 2023 202568.15

68.89

69.63

70.37

71.10Participation (- -) and trend participation rate

2017 2019 2021 2023 2025-1.12

-0.84

-0.56

-0.28

0.00Participation rate: 2020_AUTUMN_F minus 2020_SPRING_F

Participation rate Trend participation rate

2017 2019 2021 2023 20251538.98

1560.37

1581.76

1603.15

1624.55Average hours worked (- -) and trend hours worked

2017 2019 2021 2023 2025-29.89

-19.58

-9.26

1.05

11.37Average hours worked: 2020_AUTUMN_F minus 2020_SPRING_F

Average hours wkd Trend average hours wkd

2017 2019 2021 2023 20256645.29

6701.13

6756.97

6812.81

6868.65Population of working age

2017 2019 2021 2023 2025-48.67

-35.70

-22.74

-9.77

3.19Population of working age: 2020_AUTUMN_F minus 2020_SPRING_F

2017 2019 2021 2023 2025-0.04

-0.02

-0.01

0.01

0.02TFP growth (- -) and trend TFP growth

2017 2019 2021 2023 20254.42

4.84

5.25

5.66

6.08Unemployment rate (- -) and NAWRU

2017 2019 2021 2023 2025-5.75

-3.59

-1.43

0.73

2.89Output gap

2017 2019 2021 2023 2025-1.08

-0.60

-0.11

0.38

0.86Output gap: 2020_AUTUMN_F minus 2020_SPRING_F

2017 2019 2021 2023 2025342.00

354.48

366.97

379.45

391.94GDP (- -) and potential GDP

2017 2019 2021 2023 202521.85

22.71

23.56

24.42

25.28Investment to potential GDP ratio

Output gap report for AT, 2020_AUTUMN_F 2020_SPRING_F vs

EUCAM v1 (2021-04-06 10:38:56)

36

3.2.6 Worksheet Manual

The worksheet Manual contains the links to open the EUCAM and GAP manuals.

3.2.7 Worksheet Upgrades

The worksheet Upgrades allows users to easily upgrade the EUCAM software. The updated

program files upgrades.zip or upgrades.rar must be downloaded from the CIRCA web-

site and written in the folder C:\EUCAM\Upgrades. EUCAM is then updated by simply

pressing the button Upgrades (see Figure 16). The new files (with extension *.bas, *.dll,

and *.exe) overwrite the existing ones.

Figure 16 EUCAM Worksheet Program upgrades

Conversely, pressing Export saves the Visual Basic routines *.bas that are incorporated in

the EUCAM Excel file into the folder C:\EUCAM\Upgrades.

37

4 Data description

This Section describes the data involved in the EUCAM. Three sources are used: AMECO,

the DG ECFIN Business and Consumer surveys, and Eurostat.

4.1 AMECO

Most of the data come from AMECO. The AMECO database collects national accounts

data from either the national institutions, Eurostat, or the OECD. The AMECO data are

distributed in an Excel file together with EUCAM. A typical AMECO data file is illustrated

in Figure 17. Each row starting with the label ’co x’ contains data about variable ’x’ in

country ’co’. The acronyms used in the data file and their correspondence with AMECO

codes are displayed in Table A. Further details can be found in Appendix A.

38

Figure 17 Example of AMECO data file

39

Table A List of variables with correspondences

Name in AMECO Description Formula

data file

dgdpdef PVGD GDP def. % change (A3)

dlprod RVGDE Change in productivity (2.11), (A1)

dpcegdp log(Private consumption def/GDP def)

er XNE Exchange rate Output data

gdpn UVGD Nominal GDP Output data

gdpne UVGD Nominal GDP in euro Output data

gdpq OVGD Real GDP (2.14)

hpere NLHA Hours worked per employee (2.4), (2.6)

hwcdw HWCDW Nominal compensation (A3)

per employee

iq OIGT Real gross fixed capital formation (2.12)

kt OKND Capital stock (2.12)

lur ZUTN Harmonised unemployment rate (2.4), (2.10)

nulc PLCD Nominal unit labour cost

pce dpce PCPH Private consumption (A3)

expenditure deflator, % change (A3)

pgde PVGD GDP def. (euro) Output data

popa1 NPAN1 Working age population (15-74) (2.4)

popt NPTD Total population Output data

ppp KNP Purchasing power parity Output data

sle NECN Employment, civilian Output data

sle1 NETN Employment, national (2.22)

sled NETD Employment, domestic (2.7), (2.22)

winf HWCDW Wage inflation (2.11)

ws UWCD Wage share (2.11)

40

4.2 CUBS data

The CUBS series which are used to detrend TFP series are distributed in the CUBS file,

whose format is illustrated in Figure 18. CUBS is a composite indicator of capacity utili-

sation calculated using DG ECFIN data from Business and Consumer surveys. A detailed

description can be found in Annex 3 of Havik et al. (2014).

Figure 18 Example of Capacity Utilisation data file

4.3 Population

Two series, namely population of working age POPAF and total population POPTF, are

used to extend the population series. The format of the input file is illustrated in Figure 19

41

while the abbreviations are explained in Table B.

Figure 19 Example of Population data file

Table B Population variables with correspondences

Name in Eurostat Description Formula

data file

POPAF proj 18np Projection for working (2.19)

age population (15-74)

POPTF proj 18np Projection for (2.19)

total population

42

5 Conclusion

The EUCAM software implements the methodology commonly agreed between EU Member

States to estimate potential growth and the output gap. All operations are integrated in

a user-friendly environment which offers several facilities like multi-country analysis and

graphical comparisons using another data vintage. The program is open-source and can be

downloaded from CIRCABC together with the data files.

43

Appendix A AMECO data & acronyms

Compensation of employees

– UWCD: total economy.

Note: domestic concept, included are residents as well as non-residents working for

resident producer units. Compensation of employees includes wages and salaries and

employers’ social contributions.

UWCD is used for calculating the labour cost indicator (A2) or (A3) which appears in the

Phillips curve (2.11).

Consumption of fixed capital at current prices

– UKCT: total economy.

UKCT is used to measure capital.

Employment (1000 persons)

– NECN: Civilian employment.

Note: Civilian employment is equivalent to civilian labour force NLCN minus total

unemployment ZUTN. Civilian employment according to the national concept includes

residents who work for resident as well as non-resident producer units.

– NETN: national, total economy.

– NETD: all domestic industries, total economy.

Note: NETD is a domestic concept, included are residents as well as non-residents

who work for resident producer units. Employment covers employees and self-employed

persons. Annual average.

– FETD = domestic, full time equivalent, total economy.

44

– NWTD: all domestic industries excluding self-employed workers, total economy.

– FWTD: full-time equivalents; total economy.

NETN is used as number of employees when calculating labour in (2.4).

Gross domestic product at market prices

– UVGD: at current prices, total economy.

– OVGD: at constant prices, total economy.

– PVGD: price deflator calculated as 100 × UVGD/OVGD.

OVGD is used to derive the Solow residual in (2.13) and the output gap in (2.3).

Gross domestic product at market prices per hours worked

– NLHA: Average annual hours worked per person employed.

– NLHT : Total hours worked.

Hours worked per employee is one component of total labour in (2.4).

Gross domestic product at market prices per person employed

– RVGDE is calculated in constant prices either as:

RV GDE = 1000×OV GD/NETD

or = 1000×OV GD/FETD (A1)

RVGDE is used for the calculation of real unit labour costs in (A2)-(2.11).

Gross fixed capital formation at constant prices

45

– OIGT: total economy.

– PIGT: price deflator, total economy.

Net capital stock at constant prices

– OKND: total economy ( OKNDt−1+OIGTt-100 × UKCTt/PIGTt ).

Nominal unit labour cost

– PLCD: total economy.

PLCDt = 100×(UWCDt/NWTDt)

/(OV GDt/NETDt)

(UWCD95/NWTD95)/

(OV GD95/NETD95)

Nominal compensation per employee

– HWCDW: total economy, equal to either 1000×UWCD/NWTD or to 1000×UWCD/FWTD.

HWCDW is involved in the calculation of real unit labour in (A2).

Population

– NPTD: total, annual average.

– NPAN1: 15-74 years.

Working age population helps calculating the participation rate in (2.7).

Private final consumption expenditures

– UCPH: current prices.

46

– OCPH: constant prices.

– PCPH: Price deflator (100×UCPH/OCPH)

Note: private final consumption expenditure refers to the expenditure on consumption of

goods and services of households and non-profit institutions serving households. Goods

and services financed by the government and supplied to households as social transfers

in kind are not included.

PCPH is used in (A3) to obtain terms-of-trade which is one possible explanatory variable in

the backward-looking Phillips curve (2.11).

Real unit labour cost

– The growth rate of real unit labour cost is calculated as:

∆RULCt = (HWCDWt/HWCDWt−1 − 1)−∆RV GDEt − (∆ZCPIHt − 1)

(A2)

where ZCPIHt is the harmonized consumer price index.

Real unit labour cost appears as endogenous variable in the forward-looking Phillips curve

(2.11).

Terms-of-trade

– Terms-of-trade TOTt is calculated as the difference between the harmonized consumer

price index ZCPIHt and the GDP deflator PV GDt, i.e. TOTt = ZCPIHt−PV GDt.

The second-order derivative involved in (2.11) is calculated as:

ddtott =dpcet − dgdpdeft

dpcet−1 − dgdpdeft−1(A3)

where dpcet and dgdpdeft denote the first-difference of ZCPIH and PV GDt.

47

Unemployment

– ZUTN: percentage of civilian labour force; Member States, Source: AMECO.

Note: unemployed persons as a share of the labour force (number of people employed

and unemployed, see also ESA 95, paragraph 11.21). For unemployed persons see

ZUTN.

Unemployment is used to calculate the participation rate and the NAWRU in (2.7) and

(2.10).

Wage inflation

– Wage inflation WINF is calculated using net compensation of employees as in:

WINFt = HWCDWt/HWCDWt−1 − 1 (A3)

The change in wage inflation is the endogenous variable in the backward-looking wage Phillips

curve (2.11).

48

Appendix B Output files

Tables B1-B3 describe the variables produced in output. Some variables which are not

involved in the EUCAM are produced for internal needs. The output files are named for

instance like ’EUCAM output 2021 Spring F.xls’ (Table B1), ’NAWRU 2021 Spring F.xls’

(Table B2), and ’TFP 2021 Spring F.xls’ (Table B3), and can be found in the vintage folder.

49

Table B1 Variables in main output file

Y gross domestic product

Y GAP output gap estimated using EUCAM

Y Pot potential output

Part participation rate

Part S trend participation rate

wSR growth of TFP

wSR KF growth of potential TFP

NAWRU non-accelerating wage rate of unemployment

UnEmpl Rate unemployment rate

HperE hours worked per employee

Investment investment

Pot LF potential labour force

Pot Empl potential employment

Pot Tot Hrs potential total hours

Empl employment

Tot Hrs total hours

Capital capital

Wage Infl wage inflation

Civ Empl civil employment

Pop WA population of working age

Tot Pop total population

SR TFP (solow residual)

SR HP HP filtered TFP

SR Kf Potential TFP

Depreciation depreciation rate

Exchange rate exchange rate

PPS purchase power standard

GDP deflator GDP deflator

Nominal GDP-e nominal GDP in euro

NULC nominal unit labour cost

endo dwinf wage indicator used in NAWRU estimation

fitted values fitted wage Phillips curve

CUBS capacity utilisation indicator

hperehp hours worked per employee, HP-trend

whperehp growth of HP-filtered hours worked per employee

50

Table B2 NAWRU data file

NAWRU non-accelerating wage inflation rate of unemployment

endo dwinf wage indicator used in NAWRU estimation

(first difference of wage inflation)

endo drulc wage indicator used in NAWRU estimation

(first difference of real unit labour cost)

fitted values fitted wage Phillips curve

UnEmpl Rate unemployment rate

Table B3 TFP data file

cubs CUBS indicator used to detrend TFP

fitted values fitted CUBS

cu ext forecasted CUBS

sr Solow residual

tfpf forecasted Solow residual

srkf trend of Solow residual

51

Appendix C The Hodrick-Prescott filter

The Hodrick-Prescott filter decomposes an observed series xt into a trend pt plus an inde-

pendent noise ut as in xt = pt + ut by minimising the loss function:

minp1,··· ,pT

T∑t=1

u2t + λ

T∑t=3

(∆2pt)2

where ∆ denotes first-difference and λ is the inverse signal to noise ratio , for instance 1/100

for yearly data (Hodrick and Prescott, 1997). Harvey and Jaeger (1993) show that the HP

filter is a minimum mean square error filter for the I(2) plus noise model below which is thus

implicit in HP-detrending:

xt = pt + ut

∆2pt = apt λ = V (aut)/V (pt)

Appendix D The autoregressive model

In order to mitigate end effects, the HP-filter is applied on series extended with forecasts.

These forecasts are obtained fitting an autoregressive model such as:

xt = µ+ γ1xt−1 + · · ·+ γpxt−p + et

where et is a white noise. The coefficients are estimated by least-squares given the first p

observations.

52

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Publications Office of the European Union, Luxembourg, ISBN 978-92-76-20364-3, doi:10.2760/896629.

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